With regard to Globular Cluster M15, what is the difference between a core collapse and/or a black hole in the center?
Generally speaking, a Core Collapse refers to any situation where stars collide or otherwise move closer to each other. Since individual stars have massive gravity, a tightly-packed cluster of stars will attract the other stars in the cluster to it; as stars move to the center instead of remaining in their orbits, the core of the Globular Cluster gets brighter: the cluster is "collapsing" into the "core."
A Black Hole can have the same relative effect but by a different mechanism. Black holes are stars that have passed the nova-phase and collapsed into themselves, forming a ball of matter so densely packed that its gravity contains everything, even light. In this sense, if a super-massive black hole is at the center of M15, it is attracting stars to itself in greater numbers as it slowly accumulates more matter and mass. The movement of stars towards the center is, again, a "collapse" into the "core."